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Partially Penetrating Wells. By: Lauren Cameron. Introduction. Partially penetrating wells: aquifer is so thick that a fully penetrating well is impractical Increase velocity close to well and the affect is inversely related to distance from well (unless the aquifer has obvious anisotropy)

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Partially Penetrating Wells

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Partially Penetrating Wells

By: Lauren Cameron


Introduction

  • Partially penetrating wells:

    • aquifer is so thick that a fully penetrating well is impractical

    • Increase velocity close to well and the affect is inversely related to distance from well (unless the aquifer has obvious anisotropy)

      • Anisotropic aquifers

        • The affect is negligible at distances r > 2D sqrt(Kb/Kv) *standard methods cannot be used at r < 2D sqrt(Kb/Kv) unless allowances are made

  • Assumptions Violated:

    • Well is fully penetrating

    • Flow is horizontal


Corrections

  • Different types of aquifers require different modifications

    • Confined and Leaky (steady-state)- Huisman method:

      • Observed drawdowns can be corrected for partial penetration

    • Confined (unsteady-state)- Hantush method:

      • Modification of Theis Method or Jacob Method

    • Leaky (unsteady-state)-Weeks method:

      • Based on Walton and Hantush curve-fitting methods for horizontal flow

    • Unconfined (unsteady-state)- Streltsova curve-fitting or Neuman curve-fitting method

      • Fit data to curves


Confined aquifers (steady-state)

  • Huisman's correction method I

    • Equation used to correct steady-state drawdown in piezometer at r < 2D

    • (Sm)partially - (Sm)fully

      • = (Q/2∏KD) * (2D/∏d) ∑ (1/n) {sin(n∏b/D)-sin(n∏Zw/D)}cos(n∏Zw/D)K0(n∏r/D)

      • Where

        • (Sm)partially = observed steady-statedrawdown

        • (Sm)fully = steady state drawdown that would have occuarred if the wellhad been fully penetrating

        • Zw= distance from the bottom of the well screen to the underlying

        • b= distance from the top of the well screen to the underlying aquiclude

        • Z = distance from the middle of the piezometer screen to the underlying aquiclude

        • D = length of the well screen


Re: Confined aquifers (steady-state)

  • Assumptions:

    • The assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by:

      • The well does not penetrate the entire thickness of the aquifer.

  • The following conditions are added:

    • The flow to the well is in steady state;

    • r > rew

  • Remarks

    • Cannot be applied in the immediate vicinity of well where Huisman’s correction method II must be used

    • A few terms of series behind the ∑-sign will generally suffice


Huisman’s Correction Method II

  • Huisman’s correction method- applied in the immediate vicinity of well

  • Expressed by:

    • (Swm)partially – (Swm)fully = (Q/2∏D)(1-P/P)ln(εd/rew)

      • Where:

        • P = d/D = the penetration ratio

        • d = length of the well screen

        • e =l/d= amount of eccentricity

        • I = distance between the middle of the well screen and the middle of the aquifer

        • ε= function of P and e

        • rew= effective radius of the pumped well


Huisman’s Correction method II

  • Assumptions:

    • The assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by:

      • The well does not penetrate the entire thickness of the aquifer.

    • The following conditions are added:

      • The flow to the well is in a steady state;

      • r = rew.


Confined Aquifers (unsteady-state):Modified Hantush’s Method

  • Hantush’s modification of Theis method

  • For a relatively short period of pumping {t < {(2D-b-a)2(S,)}/20K, the drawdown in a piezometer at r from a partially penetrating well is

    • S = (Q/8 ∏K(b-d)) E(u,(b/r),(d/r),(a/r))

    • Where

      • E(u,(b/r),(d/r),(a/r)) = M(u,B1) – M(u,B2) + M(u,B3) – M(u,B4)

      • U = (R^2 Ss/4Kt)

      • Ss = S/D = specific storage of aquifer

      • B1 = (b+a)/r (for sympolsb,d, and a)

      • B2 = (d+a)/r

      • B3 = (b-a)/r

      • B4 = (d-a)/r


Re: Confined Aquifers (unsteady-state):Modified Hantush’s Method

  • Assumptions:- The assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by:

    • The well does not penetrate the entire thickness of the aquifer.

  • The following conditions are added:

    • The flow to the well is in an unsteady state;

    • The time of pumping is relatively short: t < {(2D-b-a)*(Ss)}/20K.


Confined Aquifers (unsteady-state):Modified Jacob’s Method

  • Hantush’s modification of the Jacob method can be used if the following assumptions and conditions are satisfied:

    • The assumptions listed at the beginning of Chapter 3, with the exception of the sixth assumption, which is replaced by:

      • The well does not penetrate the entire thickness of the aquifer.

  • The following conditions are added:

    • The flow to the well is in an unsteady state;

    • The time of pumping is relatively long: t > D2(Ss)/2K.


Leaky Aquifers (steady-state)

  • The effect of partial penetration is, as a rule, independent of vertical replenishment; therefore, Huisman correction methods I and II can also be applied to leaky aquifers if assumptions are satisfied…


Leaky Aquifers (unsteady-state):Weeks’s modification of Walton and Hantush curve-fitting method

  • Pump times (t > DS/2K):

    • Effects of partial penetration reach max value and then remain constant

  • Drawdown equation:

    • S = (Q/4 ∏KD){W(u,r/D) + Fs((r/D),(b/D),(d/D),(a/D)}

      • OR

  • S = (Q/4 ∏KD){W(u,β) + Fs((r/D),(b/D),(d/D),(a/D)}

    • Where

      • W(u,r/L) = Walton's well function for unsteady-state flow in fully penetrated leaky aquifers confined by incompressible aquitard(s) (Equation 4.6, Section 4.2.1)

      • βW(u,) = Hantush's well function for unsteady-state flow in fully penetrated leaky aquifers confined by compressible aquitard(s) (Equation 4.15, Section 4.2.3)

      • r,b,d,a = geometrical parameters given in Figure 10.2.


  • Re:Leaky Aquifers (unsteady-state):Weeks’s modification of Walton and Hantush curve-fitting methods

    • The value of f, is constant for a particular well/piezometer configuration and can be determined from Annex 8.1. With the value of Fs, known, a family of type curves of {W(u,r/L) + fs} or {W(u,p) + f,} versus I/u can be drawn

    • for different values of r/L or p. These can then be matched with the data curve for t > DS/2K to obtain the hydraulic characteristics of the aquifer.


    Re:Leaky Aquifers (unsteady-state):Weeks’s modification of Walton and Hantush curve-fitting methods

    • Assumptions:

      • The Walton curve-fitting method (Section 4.2.1) can be used if:

        • The assumptions and conditions in Section 4.2.1 are satisfied;

        • Acorrected family of type curves (W(u,r/L + fs} is used instead of W(u,r/L);

        • Equation 10.12 is used instead of Equation 4.6.

      • The Hantush curve-fitting method (Section 4.2.3) can be used if:

        • T > DS/2K

        • The assumptions and conditions in Section 4.2.3 are satisfied;

        • Acorrected family of type curves (W(u,p) + fs} is used instead of W(u,p);

        • Equation 10.13 is used instead of Equation 4.15.


    Unconfined Anisotropic Aquifers (unsteady-state):Streltsova’s curve-fitting method

    • Early-time drawdown

      • S = (Q/4∏KhD(b1/D))W(Ua,β,b1/D,b2/D)

        • Where

          • Ua = (r^2Sa)/ (4KhDt)

          • Sa = storativity of the aquifer

          • Β = (r^2/D^2)(Kv/Kh)

    • Late-time drawdown

      • S = (Q/4∏KhD(b1/D))W(Ub,β,b1/D,b2/D)

        • Where

          • Ub = (r^2 * Sy)/(4KhDt)

          • Sy = Specific yield

    • Values of both functions are given in Annex 10.3 and Annex 10.4 for a selected range of parameter values, from these values a family of type A and b curves can be drawn


    Re: Unconfined Anisotropic Aquifers (unsteady-state):Streltsova’s curve-fitting method

    • Assumptions:

    • The Streltsova curve-fitting method can be used if the following assumptions and conditionsaresatisfied:

      • The assumptions listed at the beginning of Chapter 3, with the exception of the first, third, sixth and seventh assumptions, which are replaced by

        • The aquifer is homogeneous, anisotropic, and of uniform thickness over the area influenced by the pumping test

        • The well does not penetrate the entire thickness of the aquifer;

        • The aquifer is unconfined and shows delayed watertable response.

      • The following conditions are added:

        • The flow to the well is in an unsteady state;

        • SY/SA > 10.


    Unconfined Anisotropic Aquifers (unsteady-state):Neuman’s curve-fitting method

    • Drawdown eqn:

      • S = (Q/4∏KhD)W{Ua,(or Ub),β,Sa/Sy,b/D,d/D,z/D)

        • Where

          • Ua = (r^2Sa/4KhDt)

          • Ub = (r^2Sy/4KhDt)

          • Β = (r/D)^2 * (Kv/Kh)

            • Eqnis expressed in terms of six dimensionless parameters, which makes it possible to present a sufficient number of type A and B curves to cover the range needed for field application

              • More widely applicable

              • Both limited by same assumptions and conditions


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